Application of the direct quadrature method of moments to polydisperse gas-solid fluidized beds

Most of today's computational fluid dynamics (CFD) calculations for gas-solid flows are carried out assuming that the solid phase is monodispersed, whereas it is well known that in many applications, it is characterized by a particle size distribution (PSD). In order to properly model the evolution of a polydisperse solid phase, the population balance equation (PBE) must be coupled to the continuity and momentum balance equations. In this work, the recently formulated direct quadrature method of moments (DQMOM) is implemented in a multi-fluid CFD code to simulate particle aggregation and breakage in a fluidized-bed (FB) reactor. DQMOM is implemented in the code by representing each node of the quadrature approximation as a distinct solid phase. Since in the multi-fluid model, each solid phase has its own momentum balance, the nodes of the DQMOM approximation are convected with their own velocities. This represents an important improvement with respect to the quadrature method of moments (QMOM) where the moments are tracked using an average solid velocity. Two different aggregation and breakage kernels are tested and the performance of the DQMOM approximation with different numbers of nodes are compared. These results show that the approach is very effective in modeling solid segregation and elutriation and in tracking the evolution of the PSD, even though it requires only a small number of scalars.     

A novel algorithm for solving population balance equations: the parallel parent and daughter classes. Derivation, analysis and testing

A novel numerical method, the parallel parent and daughter classes (PPDC) technique, for solving population balance equations (PBEs) is presented in this paper. In many practical applications, the PBE of particles under investigation is coupled with the thermo-fluid dynamics of the surrounding fluid. Hence, the PBE needs to be implemented in a computational fluid dynamics (CFD) code, which leads to an additional computational load. The computational cost becomes intractable when techniques such as methods of classes (CM) or Monte Carlo method are used. Quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) are accurate and require a relatively low additional computational cost when applied to CFD. The PPDC is shown to be as accurate as QMOM and DQMOM, and even more accurate in some cases, when the same number of classes is used. In the present work, the PPDC technique has been derived and tested. This technique can be used for solving a wide class of problems involving PBE such as polymerization, aerosol dynamics, bubble columns, etc. Numerical simulations have been carried out on aggregation processes with different kernels and on simultaneous aggregation and breakage processes. The numerical predictions are compared either with analytical solutions, when available, or with the numerical solutions obtained by methods of classes.     

Droplet breakage and coalescence in liquid–liquid dispersions: Comparison of different kernels with EQMOM and QMOM

Droplet coalescence and breakage in turbulent liquid–liquid dispersions is simulated by using computational fluid dynamics (CFD) and population balance modeling. The multifractal (MF) formalism that takes into account internal intermittency was here used for the first time to describe breakage and coalescence in a surfactant‐free dispersion. The log‐normal Extended Quadrature Method of Moments (EQMOM) was for the first time coupled with a CFD multiphase solver. To assess the accuracy of the model, predictions are compared with experiments and other models (i.e., Coulalogou and Tavlarides kernels and Quadrature Method of Moments [QMOM]). EQMOM and QMOM resulted in similar predictions, but EQMOM provides a continuous reconstruction of the droplet‐size distribution. Transient predictions obtained with the MF kernels result in a better agreement with the experiments. © 2016 American Institute of Chemical Engineers AIChE J, 63: 2293–2311, 2017     

Drop breakage in a single-pass through vortex-based cavitation device: Experiments and modeling

Liquid–liquid emulsions are used in many sectors such as personal care, home care, and food products. There is an increasing need for developing compact and modular devices for producing emulsions with desired droplet size distribution (DSD). In this work, we have experimentally and computationally investigated an application of vortex-based hydrodynamic cavitation (HC) device for producing emulsions. The focus is on understanding drop breakage occurring in a single-pass through the considered HC device. The experiments were performed for generating oil-in-water emulsion containing 1%–20% rapeseed oil. The effect of pressure drop across the HC device in the range of 50–250 kPa on drop breakage was examined. DSD of emulsions produced through a single pass was measured using the focussed beam reflectance measurements. Comprehensive computational fluid dynamics (CFD) model based on the Eulerian approach was developed to simulate multiphase cavitating flow. Using the simulated flow, population balance model (PBM) with appropriate breakage kernels was solved to simulate droplet breakage in a vortex-based HC device. The device showed an excellent drop breakage efficiency (nearly 1% which is much higher than other commercial devices such as rotor–stators or sonolators) and was able to reduce mean drop size from 66 to ~15 μm in a single pass. The CFD and PBM models were able to simulate DSD. The presented models and results will be useful for researchers and engineers interested in developing compact devices for producing emulsions of desired DSD.     

Implementation and analysis of numerical solution of the population balance equation in CFD packages

Simulation of polydisperse flows must include the effects of particle–particle interaction, as breakage and aggregation, coupling the population balance equation (PBE) with the multiphase modelling. In fact, the implementation of efficient and accurate new numerical techniques to solve the PBE is necessary. The direct quadrature method of moments, known as DQMOM, is a moment-based method that uses an optimal adaptive quadrature closure and came into view as a promising choice for this implementation. In the present work, DQMOM was implemented in two CFD packages: the commercial ANSYS CFX, through FORTRAN subroutines, and the open-source OpenFOAM, by directly coding the PBE solution. Transient zero-dimensional and steady one-dimensional simulations were performed in order to explore the PBE solution accuracy using several interpolation schemes. Simulation cases with dominant breakage, dominant aggregation and invariant solution (equivalent breakage and aggregation) were simulated and validated against an analytical solution. The solution of the population balance equation was then coupled to the two-fluid model, considering that all particles classes share the same velocity field. Momentum exchange terms were evaluated using the local instantaneous Sauter mean diameter of the size distribution function. The two-dimensional tests were performed in a backward facing step geometry where the vortex zones traps the particles and provides high rates of breakage and aggregation.     

Investigation of aggregation, breakage and restructuring kinetics of colloidal dispersions in turbulent flows by population balance modeling and static light scattering

Quantitative modeling of aggregating colloidal systems is the underlying problem in many industrial processes, such as micro and nanoparticle processing, crystallization or flocculation. Population balance models with various aggregation and breakage kernels have been proposed in order to describe aggregating systems, but they have been rarely validated against appropriate experimental data. Typically, model parameters are fitted against a single measured moment of the cluster distribution which can usually be equivalently described using several variations of the set of parameters underlying the relevant aggregation, breakage and restructuring kernels. In order to discriminate among alternative models we propose an approach based on measurement and quantitative modeling of multiple moments of the cluster mass distribution, such as those obtained from static light scattering measurements. This approach is applied to aggregation processes in turbulent conditions in order to test alternative kernels for aggregation, breakage, and restructuring kinetics. We present a detailed study on the sensitivity of measurables from static light scattering with respect to commonly used aggregation and breakage kinetic models. In particular, we analyze the dynamic and steady state behavior of two measurables: the average radius of gyration and the average zero angle intensity which represent two independent moments of the cluster mass distribution. In addition, we discuss the effect of cluster structure and mass distribution on the average structure factor and the apparent fractal dimension measured by static light scattering, in order to assess what structural information can be reliably extracted from such measurements.     

Modelling of flocculation using a population balance equation

In this paper, a model based on a population balance equation (PBE) is developed. It aims at reproducing experimental floc size distributions obtained at steady state in a jar-test. The objective is to develop a simple model, based on physical phenomena, and that does not contain any adjustable parameters. Floc size distributions obtained using a part of a particle image velocimetry (PIV) device and image analysis are used to develop mathematical expressions for the aggregation and breakage kernels. A critical volume beyond which breakage is of significant importance is identified and related to the hydrodynamics. Hydrodynamic sequencing allows the distribution of the daughter particles resulting from a breakage event to be established. The model is finally successfully validated against experimental results.     

On the solution of population balances for nucleation, growth, aggregation and breakage processes

This work is concerned with the modeling and simulation of population balance equations (PBEs) for combined particulate processes. In this study a PBE with simultaneous nucleation, growth, aggregation and breakage processes is considered. In order to apply the finite volume schemes (FVS) a reformulation of the original PBE is introduced. This reformulation not only help us to treat the aggregation and breakage processes in a manner similar to the growth process in the FVS but also in deriving a stable numerical scheme. Two numerical methods are proposed for the numerical approximation of the resulting reformulated PBE. The first method combines a method of characteristics (MOC) for growth process with an FVS for aggregation and breakage processes. The second method purely uses a semidiscrete FVS for all processes. Both schemes use the same FVS for aggregation and breakage processes. The numerical results of the schemes are compared with each other and with the available analytical solutions. The numerical results were found to be in good agreement with analytical solutions.     

Effect of impeller design and power consumption on crystal size distribution

Crystallization processes in a 500 mL stirred tank crystallizer with computational fluid dynamics (CFD) and population balances toward estimating how crystal size distributions (CSDs) are influenced by flow inhomogeneities was explored. The flow pattern and CSD are presented here though extensive phase Doppler particle analyzer measurements and CFD predictions for three different impeller designs (disc turbine, pitched blade turbine, and Propeller) and each rotated at three different speeds (2.5, 5, and10 r/s). As crystallization processes in practice could involve break‐up and aggregation of crystals, some selected break‐up and aggregation kernels are incorporated. Extensive comparison of simulations with experimental data showed consistent trends in the proper quantitative range. An attempt has also been made to develop scaling laws: (a) mean particle size with average power consumption per unit mass and (b) particle‐size distribution with the turbulent energy dissipation distribution. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3596–3613, 2014     

Kinetics of fluidised bed melt granulation V: Simultaneous modelling of aggregation and breakage

The main purpose of this paper is to quantify the aggregation and breakage rates in fluidised bed melt granulation (FBMG) and to subsequently relate them to various experimental conditions. The earlier paper of this series (2004d, Powder Technology 143-144, 65-83) illustrated a sequence of development and verification work on a breakage model for FBMG, based on the population balance modelling work on tracer experimental studies. A new error-weighted integral technique was also developed, which allows simultaneous extraction of the aggregation and breakage selection rate constant, as well as the attrition constant that reveals the relative amount of attrition during FBMG. Further research is conducted here, as the similar modelling strategy is employed to extract the aggregation and breakage kinetics at different operating conditions. A series of plots revealing the influence of operating conditions (binder spray rates, bed temperature, droplet size and fluidising air flow rate) on these extracted constants are therefore established. The aggregation rate constant plots reveal that the particle aggregation rate is dependent on the amount of binder available per unit time, and hence scales directly with the binder spray rate. The aggregation rate is also observed to increase with increased bed temperature when a higher viscosity binder is used, but reveals a maximum aggregation rate for a less viscous binder. The aggregation rate also increases with larger droplet size and lower fluidising air velocity. The breakage selection rate and attrition constant plot both reveal no direct dependence on binder spray rate, due to the separation in time scale over which the granule breakage occurs. The breakage rate and the extent of granule attrition is also found to decrease with increased bed temperature and increased fluidising air velocity. Due to scatter in the data, it is not possible to deduce any sensible trend on the influence of droplet size on its relative breakage rate and attrition.     

Modeling and measurement of granule attrition during pneumatic conveying in a laboratory scale system

A methodology combining theoretical and experimental techniques for characterizing and predicting the friability of granules in a laboratory scale pneumatic conveying systems is developed. Models of increasing mathematical complexity are used for analysis of experimental data. Firstly, a two-dimensional (2-D) computational fluid dynamics (CFD) model of the gas-solid flow within the Malvern Mastersizer laser diffraction equipment is developed to simulate impact of different inlet jet pressures on the flow properties and to calculate average velocity and average volume fraction of particles in the equipment. Secondly, a simple maximum-gradient population balance (MG-PB) mathematical model of breakage is developed. The model is solved using the Quadrature Method of Moments (QMOM) and used for evaluation of experimental data from the Malvern equipment. Different semi-empirical expressions for the breakage kernels and for the daughter distribution functions are tested. Multiple breakage distribution functions are needed to get satisfactory agreement with experimental data. Finally, a CFD-PB model combining CFD and QMOM methodologies is developed. The combined model employs different binary fragment distribution functions and a kernel with the breakage rate proportional to the characteristic particle size and to the square of the impact velocity between a particle and the equipment wall. Simulation results are compared with attrition experimental data indicating that the model is able to capture the qualitative trends and quantitatively predict the Sauter mean diameter d₃₂ at the outlet. However, the lower moments, in particular m₀ and m₁ are under predicted by the model. Based also on the MG-PB model results, it is our hypothesis that chipping, or breakage of particles in multiple fragments results in higher m₀ and m₁ than predicted. Further improvements of the model are proposed to incorporate multiple breakage effects. It is assumed that analogous physically based models combining properties of the gas-solid flow with the PB models can be employed to predict attrition and breakage in large-scale pneumatic conveying systems.     

Coupling of CFD with PBM for a pilot-plant tubular loop polymerization reactor

A computational fluid dynamics model, coupled with population balance model (CFD–PBM), was developed to describe the liquid–solid two-phase flow in a pilot-plant tubular loop propylene polymerization reactor. The model combines the advantage of CFD to calculate the entire flow field and that of PBM to calculate the particle size distribution (PSD). Particle growth, aggregation and breakage were taken into account to describe the evolution of the PSD. The model was first validated by comparing simulation results with the classical calculated data. Furthermore, four cases studies, involving particle aggregation, particle breakage, particle growth or involving particle growth, breakage and aggregation, were designed to identify the model. The entire flow behavior and PSD in the tubular loop reactor, i.e. PSD, solid holdup and liquid phase velocity distribution, were also obtained numerically. The results showed that the model is effective in describing the entire flow behavior and in tracking the evolution of the PSD.     

A CFD‐PBM‐PMLM integrated model for the gas–solid flow fields in fluidized bed polymerization reactors

Although the use of computational fluid dynamics (CFD) model coupled with population balance (CFD‐PBM) is becoming a common approach for simulating gas–solid flows in polydisperse fluidized bed polymerization reactors, a number of issues still remain. One major issue is the absence of modeling the growth of a single polymeric particle. In this work a polymeric multilayer model (PMLM) was applied to describe the growth of a single particle under the intraparticle transfer limitations. The PMLM was solved together with a PBM (i.e. PBM‐PMLM) to predict the dynamic evolution of particle size distribution (PSD). In addition, a CFD model based on the Eulerian‐Eulerian two‐fluid model, coupled with PBM‐PMLM (CFD‐PBM‐PMLM), has been implemented to describe the gas–solid flow field in fluidized bed polymerization reactors. The CFD‐PBM‐PMLM model has been validated by comparing simulation results with some classical experimental data. Five cases including fluid dynamics coupled purely continuous PSD, pure particle growth, pure particle aggregation, pure particle breakage, and flow dynamics coupled with all the above factors were carried out to examine the model. The results showed that the CFD‐PBM‐PMLM model describes well the behavior of the gas–solid flow fields in polydisperse fluidized bed polymerization reactors. The results also showed that the intraparticle mass transfer limitation is an important factor in affecting the reactor flow fields. © 2011 American Institute of Chemical Engineers AIChE J, 58: 1717–1732, 2012     

Analysis of breakage kernels for population balance modelling

The accurate prediction of droplet sizes is fundamental in many industrial applications. In order to be able to simulate the evolution of a size distribution, suitable kernels should be used in the population balance. In this paper we make use of four different breakage kernels in order to predict a size distribution and compare the results among them as well as with experimental data. Two breakage kernels are derived from inverse problems and the other two are derived from physical or empirical interpretations.The Sauter mean diameter, a moments-based error and Gaussian shape factors were used to compare between the resulting distributions.     

Analysis of energy dissipation in stirred suspension polymerisation reactors using computational fluid dynamics

This work evaluates the spatial distribution of normalised rates of droplet breakage and droplet coalescence in liquid–liquid dispersions maintained in agitated tanks at operation conditions normally used to perform suspension polymerisation reactions. Particularly, simulations are performed with multiphase computational fluid dynamics (CFD) models to represent the flow field in liquid–liquid styrene suspension polymerisation reactors for the first time. CFD tools are used first to compute the spatial distribution of the turbulent energy dissipation rates (ε) inside the reaction vessel; afterwards, normalised rates of droplet breakage and particle coalescence are computed as functions of ε. Surprisingly, multiphase simulations showed that the rates of energy dissipation can be very high near the free vortex surfaces, which has been completely neglected in previous works. The obtained results indicate the existence of extremely large energy dissipation gradients inside the vessel, so that particle breakage occurs primarily in very small regions that surround the impeller and the free vortex surface, while particle coalescence takes place in the liquid bulk. As a consequence, particle breakage should be regarded as an independent source term or a boundary phenomenon. Based on the obtained results, it can be very difficult to justify the use of isotropic assumptions to formulate particle population balances in similar systems, even when multiple compartment models are used to describe the fluid dynamic behaviour of the agitated vessel. © 2011 Canadian Society for Chemical Engineering     

Fluid Simulation-Supported Extraction Process Design: An Approach Towards Improving Current Models

In recent years, computational fluid dynamics (CFD) has become a valuable tool for the design of extraction processes. CFD simulations have been combined with population balances to account for coalescence and breakage. In this work, another approach to account for drop interactions is shown, namely, the incompressible Cahn-Hilliard/Navier-Stokes equations. To apply the Cahn-Hilliard equations, they have to be combined with a thermodynamic model. Here, a Koningsveld-Kleintjens approach is used to account for the mixing gap of the binary mixture. The suggested method was applied to analyze drops in a shear field and to investigate drop coalescence.     

Breakage model development and application with CFD for predicting breakage of whey protein precipitate particles

Particle breakage due to fluid flow through various geometries can have a major influence on the performance of particle/fluid processes and on the product quality characteristics of particle/fluid products. In this study, whey protein precipitate dispersions were used as a case study to investigate the effect of flow intensity and exposure time on the breakage of these precipitate particles. Computational fluid dynamic (CFD) simulations were performed to evaluate the turbulent eddy dissipation rate (TED) and associated exposure time along various flow geometries. The focus of this work is on the predictive modelling of particle breakage in particle/fluid systems. A number of breakage models were developed to relate TED and exposure time to particle breakage. The suitability of these breakage models was evaluated for their ability to predict the experimentally determined breakage of the whey protein precipitate particles. A “power-law threshold” breakage model was found to provide a satisfactory capability for predicting the breakage of the whey protein precipitate particles. The whey protein precipitate dispersions were propelled through a number of different geometries such as bends, tees and elbows, and the model accurately predicted the mean particle size attained after flow through these geometries.     

Simulation of the population balances for liquid-liquid systems in a nonideal stirred tank. Part 2—parameter fitting and the use of the multiblock model for dense dispersions

In the previous part of this work (Chem. Eng. Sci. 54 (1999) 5887), a multiblock simulation model was developed in order to allow the close examination of different regions of a stirred tank for drop size distribution calculations. In this paper, that model is tested in a parameter fitting procedure. The drop breakage and coalescence parameters are fitted against drop size measurements from dense liquid-liquid dispersions, which were assumed fully turbulent. Since the local turbulence and flow values of a stirred tank are used in the present model, the fundamental breakage and coalescence phenomena can be examined more closely. Furthermore, the present model is capable of predicting inhomogeneities occurring in a stirred tank. It is also to be considered as an improved tool for process scale-up, compared to the simple vessel-averaged population balance approach, or use of correlations of dimensionless numbers only. The present model can use two sources of data for fitting parameters in the drop rate functions. One is to use transient data of the measured drop size distribution as the impeller speed is changed. The other is to use time-averaged data measured at different locations of the stirred tank. It is shown in this paper that the different flow regions can be chosen from the CFD simulations in a straightforward manner. CFD flow simulation results can be used to select the flow regions when no experimentally obtained flow conditions are available. This is especially useful for non-standard vessels, such as reactors containing cooling coils. After fitting the parameters with a multiblock model, the population balance model can be rather easily incorporated into a commercial CFD program to investigate different flow conditions.